Multilevel methods for mixed finite elements in three dimensions

In this paper we consider second order scalar elliptic boundary value probl ems posed over three-dimensional domains and their discretization by means of mixed Raviart-Thomas finite elements [18]. This leads to saddle point pr oblems featuring a discrete flux vector field as additional unknown. Following Ewing and Wang [26], the proposed solution procedure is based on splitting the flux into divergence free components and a remainder. It lead s to a variational problem involving solenoidal Raviart-Thomas vector field s, A fast iterative solution method for this problem is presented. It exploits the representation of divergence free vector fields as curls of the H(curl )conforming finite element functions introduced by Nedelec [43]. We show th at a nodal multilevel splitting of these finite element spaces gives rise t o an optimal preconditioner for the solenoidal variational problem: Duality techniques in quotient spaces and modern algebraic multigrid theory [50, 1 0,31] are the main tools for the proof.